Long-term control of HIV depends on improvement in an individual's immune system. The purpose of this study is to see if either stopping anti-HIV drugs for short periods of time and/or adding a vaccine to the anti-HIV drugs being taken will help to better control HIV infection. The study will test whether these treatment approaches are safe. The HIV vaccine in this study has been tested in people who did not have HIV infection and improved the way their immune system worked. This study will evaluate whether these same immune system changes happen in people with HIV, and, if such changes do occur, assess whether these changes help to improve control of HIV in these patients.